1. Field of the Invention
This invention pertains in general to the field of phase-shifting interferometry and, in particular, to a novel approach for measuring thin-film thickness and step heights between regions of a surface having different phase change on reflection.
2. Description of the Related Art
Optical surface profilers based on phase-shifting interferometry (PSI) utilize phase measurements to calculate the surface height values, h(x,y), at each point of a surface under test. The technique is founded on the concept of varying the phase difference between two coherent interfering beams of single wavelength in some known manner, such as by changing the optical path difference (OPD) in discrete steps or linearly with time. Under such conditions, three or more measurements of the light intensity at a pixel of a receiving sensor array can be used to determine the initial phase difference of the light beams at the point on the test surface corresponding to that pixel. Based on such measurements at each pixel of coordinates x and y, a phase distribution map .PHI.(x,y) can be determined for the test surface, from which very accurate height data h(x,y) are calculated by the following general equation in relation to the wavelength .lambda. of the light source used: ##EQU1## Phase-shifting interferometry provides a vertical resolution on the order of 1/100 of a wavelength or better and is widely used for measuring opaque surfaces of similar materials. However, existing techniques for reconstructing surface profiles from phase measurements are inadequate for samples having optically-thin films over opaque substrates because the phase change associated with the beam reflected from the thin-film surface is distorted by interference between the light reflected from the top and the bottom of the film. An optically-thin film is defined as having a film thickness that is less than the coherence length of the light source, or less than the skin depth of the material, or less than the depth of field of the interferometric microscope. Also, if the substrate material is not optically dielectric (i.e., the extinction coefficient of the material is not zero), a phase change occurs on reflection from the substrate as well (referred to in the art as Fresnel phase change on reflection). In such cases the phase changes on reflection from the film and from the substrate (or from regions of different thin-film thickness) vary with several parameters including the film thickness and the optical constants of the materials composing the film and the substrate. Therefore, unless these parameters are all known, a correct profile measurement of a thin-film/substrate surface is impossible with conventional techniques, which are limited to measurements of homogeneous surfaces.
The relationship between the phase change associated with a beam reflected at the interface between an incident medium (such as air) and a film assembly (film plus substrate) and the physical properties of the media are well understood in the art. For example, referring to the case illustrated in FIG. 1, where an optically-thin film 10 of thickness t (which is also the step height h) is deposited over a portion of an opaque substrate 12, the phase change of a normal incident beam L (shown incident at an angle to illustrate reflection according to acceptable convention) reflected from the substrate 12 can be calculated by the following equation ##EQU2## where n.sub.s and k.sub.s are the refraction index and the extinction coefficient of the substrate material, respectively. It is known that the substrate parameters n.sub.s and k.sub.s vary with the wavelength .lambda. of the incident light; therefore, Equation 2 can be expressed as .DELTA..phi.=.DELTA..phi.(.lambda.). Accordingly, the phase change .DELTA..phi. of a normal incident light of wavelength .lambda. reflected from the substrate 12 can be calculated exactly if the refraction index and extinction coefficient at that wavelength are know.
Similarly, the phase change associated with the reflection of the same beam L normally incident to the thin film 10 from air can be obtained by the equation ##EQU3## where .eta..sub.o is the admittance of air, and a and b are the real and imaginary components, respectively, of the input optical admittance Y of the assembly consisting of the thin film 10 and the substrate 12 (i.e., Y=a+ib). As those skilled in the art readily understand, the admittance Y is a function of the thickness t and the optical constants of the thin film 10 and the optical constants of the substrate 12 through a complex relationship that is omitted here for simplicity. For details, see H. A. Macleod, "Thin-Film Optical Filters," 2nd Edition, McGraw Hill, New York (1989), pp. 34-37. Since the optical constants of the film 10 and substrate 10 also vary with the wavelength .lambda. of the incident light L, Equation (3) can be represented by the expression .DELTA..phi.=.DELTA..phi.(.lambda.,t). Therefore, the phase change .DELTA..phi. of a normal incident light of wavelength .lambda. reflected from the thin film 10 can be calculated exactly from the admittance of the assembly if the thickness t of the film (which is also the height h of the step) and the optical constants of the media at that wavelength are know.
In general, when a surface having a step discontinuity h, such as that provided by the thickness t of film 10 in the model of FIG. 1, is scanned to measure the height h by phase-shifting interferometry with a light beam of wavelength .lambda., it is known that the measured height, h.sub.m, will differ from the true (physical) height, h, according to the relationship ##EQU4## where .DELTA..phi..sub.1 and .DELTA..phi..sub.2 represent, respectively, the phase changes on reflection from the interface between air and the first region (i.e., the film in the example of FIG. 1) and air and the second, stepped region (i.e., the substrate). From Equations 2 and 3, it is clear that .DELTA..phi..sub.1 =.DELTA..phi..sub.1 (.lambda.,t) and .DELTA..phi..sub.2 =.DELTA..phi..sub.2 (.lambda.) for the model of FIG. 1. Therefore, the relationship between the true and measured step heights for the case where the first region consists of a uniform thin film and the second region consists of an opaque substrate (that is, t=h) becomes ##EQU5## It is noted that Equation 5 represents an implicit relationship of h as a function of h.sub.m. Thus, the equation cannot be solved directly to calculate the true step height h as a function of h.sub.m once a measured height h.sub.m (.lambda.) is determined by phase-shifting interferometry at a given wavelength .lambda.. In addition, the functionality changes with the wavelength of the incident light because of the different values assumed by the various optical constants of the materials (n, k, .eta..sub.o, Y) as the wavelength varies. Therefore, phase-shifting measurements conducted with two different wavelengths, .lambda..sub.1 and .lambda..sub.2, on a sample with a physical film of thickness h will produce two different measured heights, h.sub.m1 and h.sub.m2, respectively. As detailed below, this invention exploits this property to solve Equation 5 explicitly for h (and therefore t) by a numerical approach.
The terms step height and film thickness are used throughout this disclosure to describe the height difference between adjacent regions of a test surface and the thickness of a thin film overlaying a substrate, respectively. The two are obviously the same and can be used interchangeably in the case of a uniform thin film covering a portion of a substrate, such as seen in FIG. 1, but refer to different physical structures in the case of a non-uniform stepped film, as illustrated in FIG. 2, or a uniform film over a stepped substrate (that is, a substrate containing a riser), as illustrated in FIG. 3. In all cases, however, the above-referenced Equations 2-4 from thin-film theory provide the relationships necessary to investigate the phase changes on reflection and their effect on step height measurements by phase shifting. In addition, the general approach of this invention can be extended to produce accurate surface profiles of thin films irrespective of the particular structure of discontinuities.
Prior art techniques, such as described in U.S. Pat. Nos. 5,129,724 and 5,173,746, utilize the above-described relationships to numerically calculate the true height of a step (or thickness of a thin film) from phase-shifting measurements by iterative procedures that are time consuming and require a substantially accurate initial guess that necessitates additional measurements than required for conventional phase-shifting procedures. In particular, these techniques are only suitable for the case shown in FIG. 1; that is, when the sample consists of two regions, a uniform film and a substrate. Even for this case, the invention only works when the measured step height h.sub.m is a monotonic function of the film thickness t. Otherwise, depending on the initial guess, the procedure may produce a wrong solution. In practice, such monotonic relationship hold true for very few materials. These aspects of these techniques are problematic because range and speed of measurement are critical requirements for commercial instruments used for quality control during production.
The present invention is directed at providing an approach that greatly improves prior-art techniques with respect to these problems by allowing a direct determination of the step height h once phase-shifting measurements are carried out. By eliminating the need for a suitable initial guess, it more reliably provides accurate results. In addition, it is founded on a general approach applicable to reconstruct 3-D profiles of samples consisting of different film and substrate combinations.